$$ \int (a \psi + b ) d\psi= a \hspace{0.333in}\text{and}\hspace{0.333in} \int e^{\phi^T A \psi}\; d\phi d\psi = \det A$$
I am not sure what the analog is for 1-forms in the cotangent bundle of a manifold $T^1(\mathbb{R}^n)$.

It also seems these kinds of "free-fermion" calculations lead to many generalizations of matrix-tree that I won't get into (including relations to Branched Polymers and Diffusion Limited Aggregation). Personally, I wonder what the cotangent bundle picture looks like for these.

The proofs in the two above papers are left as exercises, which more general results proven. Mainly, I just would like to see the proofs the no-frills Matrix-Tree theorem from either of these two starting points